Prove (square root 2) is irrational - I have a problem with this proof.

root2 = a/b where a and b are both even therefore a and b are not coprime which contradicts the initial condition that a and b must be coprime therefore root2 is irrational.
-------------------------------------------------
I ran through a different proof to attempt to prove that root4 is irrational
(this is not the full proof, I have truncated it a little)
suppose root4 were irrational
then root4 = a/b
4b^2=a^2
therefore a is even
Let a=2k
4b^2=4k^2
b^2=k^2
k=+-b
therefore
a=2k and b=+/-k
Therefore a and b are not coprime
Therefore root4 is irrational.

Obviously I realize that 2k/k can be reduced to 2/1 which then shows root4 to be rational

BUT

when the first proof was looking at root2 and it got down to (an even number/an even number) so why is it impossible for that fraction also be reduced to some p/q where p and q are coprime. To me, the proof just doesn't seem to be finished and does not prove anything (not to me).

Re: Prove (square root 2) is irrational - I have a problem with this proof.

root2 = a/b where a and b are both even therefore a and b are not coprime which contradicts the initial condition that a and b must be coprime therefore root2 is irrational.
-------------------------------------------------
I ran through a different proof to attempt to prove that root4 is irrational
(this is not the full proof, I have truncated it a little)
suppose root4 were irrational
then root4 = a/b
4b^2=a^2
therefore a is even
Let a=2k
4b^2=4k^2
b^2=k^2
k=+-b
therefore
a=2k and b=+/-k
Therefore a and b are not coprime

Re: Prove (square root 2) is irrational - I have a problem with this proof.

We don't know exactly what k is although it does have a specific value. We do know that 'a' has a factor of 2 so therefore 'b' cannot be even.
etc
--------------------------------------------------
If I make this change to the wording of the proof then I am happy and I think it all makes sense.

Thankyou for your help Romsec. I have enjoyed our communications this afternoon and I appreciate your help.
----------------------------------------------------

I just a had a brainwave. Of course k has a specific value because 'a' has to have a specific value and k is half of a.
Now I have it. In the sqrt4 proof k has to be 1, there is not other possibliliy.
Thankyou so much for helping me reach this understanding.
---------------------------------------------------------
I know you kept telling me that k=1 but the real reason for this was not sinking into my brain.
Sometimes students (that's me) can be very frustrating. I didn't mean any offence by the comment.